Various Math #68 - Q&A

Question 1 ABC Company manufactures purple widgets. The company has established a total standard cost for direct material and direct labor of $244.33 per widget. The total standard cost includes 21.5 pounds of direct materials at a standard cost of $10.50 per pound and 3.45 hours of direct labor at a wage rate of $8.50 per hour. The following data pertain to April: The company purchased 500,000 pounds of material, 90% of the material purchased was used for direct materials. The total cost of the direct materials was $4,050,000. The total amount paid for direct labor was $600,750. The actual direct labor hours worked was 67,500 hours. The company actually produced 20,000 widgets for the month of April a) Calculate the Total Direct Materials Variance b) Calculate the Total Direct Labor Variance c) Explain who is in the best position in the organization to influence each of these variances. Direct Materials Price Variance Direct Materials Quantity Variance Direct Labor Rate Variance Direct Labor Time Variance Answer Step 1 Direct Materials cost variance = Standard cost of materials - Actual cost of materials = (21.5 x 10.5 x 20,000) - (4,050,000 × 90%) = 4515,000 - 3645000 = $870,000(Favourable) Direct Labour cost variance = Standard cost of labour - Actual cost of labour = (3.45 x 8.5 × 20,000) – 600, 750 = $11250(Unfavourable) Step 2 Unfavourable direct materials price variance means variance in standard cost and actual cost due to standard price is less than actual price of materials Favorable Direct Materials Quantity Variance means variance in standard cost and actual cost due to standard quantity used is more than actual quantity cost. Favourable total direct materials cost variance in standard cost and actual cost when standard costs of materials are more than actual cost of materials. Favorable Direct Labor Rate Variance means variance in standard labour cost and actual labour cost when actual labour rate is less than standard rate. Unfavorable Direct Labor Time Variance means variance in standard labour cost and actual labour cost when actual labour hours are more than standard labour hours. Unfavorable Total Direct Labor Variance means variance in standard labour cost and actual labour cost when actual cost of labour is more than standard labour cost. Question 2 Write in simplest radical form: a) 2√200 - √98 b) ∛32x³ - 2√x + 2∛25 c) ∛(2x²)/y Answer 1 a) 2√200 - √98 2 * 10√2 - 7√2 = 13√2 Answer 2 b) ∛32x³ - 2√x + 2∛25 = 4x∛2x - 2√x + 10 = 2x∛4x - 2√x + 10 c) ∛(3x²/y) = ∛(3x²y²)/y³ = ∛(3x²y²) / y Question 3 Two in-phase loudspeakers emit identical 1000 Hz sound waves along the x-axis. What distance should one speaker be placed behind the other for the sound to have an amplitude 1.20 times that of each speaker alone? Answer Solution: Consider the formula to findout the wavelength is, λ= v/f Substitute 343 m/s for v and 1000Hz for f in equation λ = 343 m/s / 1000 Hz = 0.343 m Consider the formula to find the amplitude of the wave formed due to two waves with same frequency A = 2acos(Δφ /2) Here, Δφ = 2πΔx/λ Therefore, A = 2acos((2πΔx/λ) / 2) Substitute (1.20 × a) for A and 0.343 m for λ in equation 1.20a = 2acos((2πΔx / 0.343 m) / 2) Δx = (0.343 m / π) * cos⁻¹(1.20/2) = 0.0579 Δx = 5.79 cm Thus, the distance that one speaker be placed behind the other for the sound to have an amplitude 1.20 times that of each speaker alone is Δx = 5.79 cm Question 4 The average distance between Earth and the Sun is 1.5 x 10¹¹m. (a) Calculate the average speed of Earth in its orbit (assumed to be circular) in meters per second. (b) What is this speed in miles per hour? Answer We know the average distance [dg] from the Earth to the Sun = 1.5 × 10¹¹m The travel distance d₁ is given: d₁ = 2π× dg = 2π * 1.5 × 10¹¹ = 9.424 × 10¹¹m And v₁ is given by: v1 = d₁ / t v1 = 9.424 × 10¹¹m / 1year = 9.424 × 10¹¹m / (365.25day × 24 × 3600s) ≈ 3 × 10⁴ m/s The speed of Earth on average = 29862.85 m/s Speed is given by: V₂ = 29862.85 m/s × (3600 h / 1609.34 mi) = 6.7 × 10⁴ m/h Therefore, (a) v₁ ≈ 3 × 10⁴ m/s , (b) v₂ ≈ 6.7 × 10⁴ m/h Question 5 A quarterback takes the ball from the line of scrimmage, runs backwards for 10.0 yards, then runs sideways parallel to the line of scrimmage for 15.0 yards. At this point, he throws a 50.0-yard forward pass straight downfield, perpendicular to the line of scrimmage. What is the magnitude of the football's resultant displacement? Answer Let us denote the displacements of the ball made by the quarterback as: ΔR₁, ΔR₂ and ΔR₃ . These are: running backwards for 10 yards, running parallel to the line of scrimmage, sideways, for 15 yards and throwing the ball 50 yards forward respectively. We can calculate the resulting displacement, ΔR by adding all of the displacements. The order in which we do it shouldn't